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PROPOSITION XII. THEOREM.

69. CONVERSELY: When two straight lines are cut by a third straight line, if the alternate-interior angles be equal, the two straight lines are parallel.

M ...............

Let E F cut the straight lines A B and C D in the points

H and K, and let the Z A H K = L HKD.

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then

§ 68

Through the point H draw MN || to CD;

Z MHK = L HKD,

(being alt.-int. £).
ZAH K = L HKD,

iiZ MHK = L A H K.
.. the lines M N and A B coincide.

But

Нур.

Ax. 1.

But

M N is || to CD;

Cons.

.. A B, which coincides with M N, is || to C D.

Q. E. D.

PROPOSITION XIII. THEOREM. 70. If two parallel lines be cut by a third straight line, the exterior-interior angles are equal.

A

Let A B and CD be two parallel lines cut by the

straight line E F, in the points H and K.

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71. COROLLARY. The alternate-exterior angles, EH B and CKF, and also A H E and D KF, are equal.

PROPOSITION XIV. THEOREM.

72. CONVERSELY: When two straight lines are cut by a third straight line, if the exterior-interior angles be equal, these two straight lines are parallel.

Let EF cut the straight lines A B and C D in the

points H and K, and let the < EHB= 2 H K D.

We are to prove AB || to CD.
Through the point H draw the straight line M N || to C D.
Then ZEHN= _ HKD,

§ 70
(being ext.-int. 1).
But
ZEHB= 2 H K D.

Hyp. .. ZEHB= 2 EHN.

Ax. 1.

.. the lines M N and A B coincide.
But

M N is || to CD,
... A B, which coincides with M N, is | to C D.

Cons.

Q. E. D.

Proposition XV. THEOREM.

73. If two parallel lines be cut by a third straight line, the sum of the two interior angles on the same side of the secant line is equal to two right angles.

Let A B and C D be two parallel lines cut by the

straight line EF in the points II and K.
We are to prove 2 BHK + L HKD = two rt. .

Z EHB + Z B H K = 2 rt. 4, § 34

(being sup.-adj. €).
Z EHB= 2 II K D,

§ 70 (being ext.-int. 1). Substitute 2 HKD for 2 E II B in the first equality ; then

ZBH K + HKD = 2 rt. s.

But

Q. E. D.

PROPOSITION XVI. THEOREM. 74. CONVERSELY: When two straight lines are cut by a third straight line, if the two interior angles on the same side of the secant line be together equal to two right angles, then the two straight lines are parallel.

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Let EF cut the straight lines A B and C D in the

points H and K, and let the Z B H K + Z HKD equal two right angles.

We are to prove A B |to C D.
Through the point H draw M N || to C D.
Then NHK + 2 HKD = 2 rt. I,

$ 73
(being two interior on the same side of the secant line).
But LBHK + Z HKD = 2 rt. L. Hyp.
..ZNH K+ ZH KD=2BHK+Z H K D. Ax. 1.
Take away from each of these equals the common Z II KD,
then

NHK = LBH K.
is the lines A B and M N coincide.
But
M N is |to C D;

Cons. .. A B, which coincides with M N, is || to C D.

Q. E D.

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